# PECOK: a convex optimization approach to variable clustering

Abstract : The problem of variable clustering is that of grouping similar components of a $p$-dimensional vector $X=(X_{1},\ldots,X_{p})$, and estimating these groups from $n$ independent copies of $X$. When cluster similarity is defined via $G$-latent models, in which groups of $X$-variables have a common latent generator, and groups are relative to a partition $G$ of the index set $\{1, \ldots, p\}$, the most natural clustering strategy is $K$-means. We explain why this strategy cannot lead to perfect cluster recovery and offer a correction, based on semi-definite programing, that can be viewed as a penalized convex relaxation of $K$-means (PECOK). We introduce a cluster separation measure tailored to $G$-latent models, and derive its minimax lower bound for perfect cluster recovery. The clusters estimated by PECOK are shown to recover $G$ at a near minimax optimal cluster separation rate, a result that holds true even if $K$, the number of clusters, is estimated adaptively from the data. We compare PECOK with appropriate corrections of spectral clustering-type procedures, and show that the former outperforms the latter for perfect cluster recovery of minimally separated clusters.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.inrae.fr/hal-02966884
Contributor : Nicolas Verzelen <>
Submitted on : Wednesday, October 14, 2020 - 3:02:56 PM
Last modification on : Friday, June 11, 2021 - 5:12:08 PM

### Identifiers

• HAL Id : hal-02966884, version 1
• ARXIV : 1606.05100

### Citation

Florentina Bunea, Christophe Giraud, Martin Royer, Nicolas Verzelen. PECOK: a convex optimization approach to variable clustering. 2020. ⟨hal-02966884⟩

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